Jacobi identity electrodynamics

The homogeneous field equation of 0(3) electrodynamics is inferred from the Jacobi identity of variant derivatives... 

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The E(3) field is zero in frame K, and a Z boost means [from Eq. (403)] that it is zero in frame K. This is consistent with the fact that Ea> is a solution of an invariant equation, the Jacobi identity (30) of 0(3) electrodynamics. Finally, we can consider two further illustrative example boosts of El ]] in the X and Y directions, which both produce the following result ... 

The field equations of electrodynamics for any gauge group are obtained from the Jacobi identity of Poincare group generators [42,46] ... 

The Jacobi identity (40) means that the homogeneous field equation of electrodynamics for any gauge group is... 

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